Method of systematic construction of space-time constellations, system and method of transmitting space-time constellations

ABSTRACT

Space-time code, and methods for constructing space-time codes are provided. The space-time coder performs a respective linear transformation on each of P sets of K modulated symbols of a modulated symbol stream to produce P sets of T linearly transformed symbols, applies a respective phase rotation to each set of T linearly transformed symbols to produce a respective set of T phase rotated symbols, and performs a threading operation on the sets of T phase rotated symbols to produce P threaded sequences that define M output sequences. During each of T symbol periods, a respective one of the P threaded sequences includes a symbol from one of the P sets of phase rotated symbols. At least one symbol from each set of phase rotated symbols appears in each output sequence, where M&gt;=2, 2&lt;=P&lt;=M, and T&gt;=M and M&gt;=K.

RELATED APPLICATION

This application is a continuation of U.S. application Ser. No.11/095,430, filed Mar. 31, 2005, now U.S. Pat. No. 7,583,747 whichclaims the benefit of U.S. Provisional Application 60/558,263 filed Mar.31, 2004, both of which are hereby incorporated herein in theirentireties.

FIELD OF THE INVENTION

The invention relates to space-time constellations for use in MIMO(multiple-input, multiple-output) communications systems.

BACKGROUND OF THE INVENTION

Wireless channels are characterized by complex physical layer effectsresulting from multiple users sharing spectrum in a multipath fadingenvironment. In such environments, reliable communication is sometimespossible only through the use of diversity techniques in which thereceiver processes multiple replicas of the transmitted signal undervarying channel conditions. Antenna diversity techniques have receivedconsiderable attention recently due to the significant gains promised byinformation-theoretic studies. While the use of multiple receiveantennas is a well-explored problem, the design of space-time (ST)signals that exploit the available capacity in multi-transmit antenna ormulti-transmit/multi-receive antenna systems still faces manychallenges. Tarokh et al. coined the name space-time coding for thistwo-dimensional signal design paradigm. Over the past five years,several ST coding schemes have been proposed in the literature.

The design of full diversity algebraic constellations for theRayleigh-fading channel was pioneered by Boullé and Belfiore. The mainidea behind their work is to introduce redundancy in the signal space(or signal space diversity) when the signal constellation is carved fromsome algebraic lattices. Signal space diversity can be obtained byapplying fully diverse unitary transformations to inputs drawn fromlattices or multidimensional digital modulation signals carved from anumber ring, such as pulse amplitude modulation (PAM) or quadratureamplitude modulation (QAM) constellations. The resulting constellationshave the property that each point is uniquely determined by any of itscomponents which allows for the possibility of retrieving the wholepoint if some of the components are lost in a deep fade.

SUMMARY OF THE INVENTION

According to one broad aspect, the invention provides an apparatuscomprising: a space-time coder adapted to perform a respective lineartransformation on each of P sets of K modulated symbols of a modulatedsymbol stream to produce P sets of T linearly transformed symbols, toapply a respective phase rotation to each set of T linearly transformedsymbols to produce a respective set of T phase rotated symbols, and toperform a threading operation on the sets of T phase rotated symbols toproduce P threaded sequences that define M output sequences; thethreading operation being such that each threaded sequence is anallocation of output sequences over time of a respective one of the Psets of T phase rotated symbols in which all of the output sequences areused by each threaded sequence; during each of T symbol periods, arespective one of the P threaded sequences includes a symbol from one ofthe P sets of phase rotated symbols; and at least one symbol from eachset of phase rotated symbols appears in each output sequence; whereM>=2, 2<=P<=M, and T>=M and M>=K.

In some embodiments, the apparatus further comprises: M transmitantennas; wherein each output sequence is transmit from a respective oneof the M transmit antennas.

In some embodiments, each antenna is used equally often to the extentpossible having regard to the number of antennas M and the number ofsymbols T in each threaded sequence.

In some embodiments, the apparatus further comprises: a channelcoder/modulator adapted to produce the modulated symbol stream from aninput information stream.

In some embodiments, the linear transformations are all identical andunitary.

In some embodiments, the linear transformations comprise at least inpart a DFT (discrete Fourier transform).

In some embodiments, each linear transformation is the product of a DFTand a diagonal matrix.

In some embodiments, each linear transformation is one of the optimizedlinear transformations defined by one or more of equations 10a, 10b, 10cand 10d.

In some embodiments, the phase rotations comprise φ_(i)=φ^(i−1) i=1, M,for some value φ.

In some embodiments, φ is as defined in Theorem 3.

In some embodiments, the space-time code is completely specified by theparameters M (the number of transmitter antennas), N (the number ofreceive antennas), T (the number of symbol periods, i.e. the codeconsists of M×T signalling matrices), P (the number of threads givingthe rate of the code in symbols from the input alphabet per channel use;we set P=min (M,N) in order to achieve full rate), θ (the Diophantinenumber within each thread in the M×T space-time matrix) and φ (theDiophantine number between the threads).

In some embodiments, the parameters are in accordance with one of thesets in the following table:

TABLE M N T P θ φ 2 1 2 1 exp(iπ/4) 1 3 1 3 1 exp(2π/5) 1 4 1 4 1exp(iπ/8) 1 2 2 2 2 exp(iπ/4) exp(iπ/6) 4 2 4 2 exp(iπ/8) exp(iπ/6) 3 33 3 exp(2π/5) exp(2π/7) 4 4 4 4 exp(iπ/8) exp(iπ/32) 6 6 6 6 exp(2π/7)exp(2π/42) 8 4 8 4 exp(iπ/16) exp(iπ/64) 10 5 10 5 exp(2π/11) exp(2π/55)12 12 12 12 exp(2π/13) exp(2π/156)

In some embodiments, an apparatus is adapted to transmit a space-timecode defined by equation 27 for a MISO channel.

In some embodiments, an apparatus is adapted to transmit a space-timecode defined by one of equations 32, 33, 34, 35, 36 for MIMO channels.

In some embodiments, the apparatus is applied in a collaborative schemein a wireless network.

In some embodiments, a storage and retrieval system comprising theapparatus as summarized above.

In some embodiments, an OFDM transmitter comprising the space-time coderas summarized above, wherein respective transmit sequences aretransmitted on respective sub-carriers.

In some embodiments, an apparatus comprising a space-time coderequivalent to the space-time coder as summarized above.

According to another broad aspect, the invention provides a method ofconstructing a full diversity algebraic constellation for use with thestructure of any preceding claim, the method comprising: selecting aparameterized linear transformation; performing an optimization ornear-optimization of the parameterized linear transformation; performinga further optimization to determine phase rotations φ_(i), for i=0, . .. , M−1.

In some embodiments, the parameterized linear transformation comprises aDFT multiplied by a diagonal matrix.

In some embodiments, the parameterized linear transformation is asdefined in equation 10a.

In some embodiments, parameters include φ₁, φ₂, . . . , φ_(M) used inthe diagonal matrix diag (φ₁, φ₂, . . . , φ_(M)).

In some embodiments, the diagonal matrix is defined as a function of asingle parameter φ.

BRIEF DESCRIPTION OF THE DRAWINGS

Preferred embodiments of the invention will now be described withreference to the attached drawings in which:

FIG. 1 is a block diagram of a space-time coding system provided by anembodiment of the invention;

FIG. 2 is a flowchart of a method of constructing a constellationprovided by an embodiment of the invention;

FIG. 3 is a block diagram of another space-time coding system providedby an embodiment of the invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS Coding SystemStructure

Referring now to FIG. 1, shown is a block diagram of a space-time codingsystem provided by an embodiment of the invention. Data symbols (bits orotherwise) are input to a channel coding/modulation function 10 toproduce modulated symbols 12 of an appropriate symbol alphabet. Examplesof suitable alphabets include but are not limited to quadratureamplitude modulation, pulse amplitude modulation and phase shift keying.The system is an M×N MIMO system, meaning there are M transmit antennas,and N receive antennas. The M transmit antennas are indicated at 14. Thereceive antennas form part of a receiver (not shown). Sets of P×Ksymbols are processed at once by the functionality which follows thechannel coding/modulation 10.

K symbols are input to each of P linear transformations 16 to produce arespective set 18 of T linearly transformed symbols, where T≧K. Inpreferred embodiments, the linear transformations 18 are all identicaland unitary. Thus, a total of K×P symbols are processed at a time toproduce K×T symbols. In a preferred embodiment, P (=number of lineartransformations)=M (=number of antennas).

In a preferred embodiment, the linear transformation comprises at leastin part a DFT (discrete Fourier transform). The effect of the lineartransformation is to introduce correlation between each of the symbolsoutput from the linear transformation. Other examples include repetitioncodes, and a combined unitary transformation and repetition code.Preferably, the linear transformation is selected to achieve diversitywhile ensuring linearity.

Preferably, the linear transformation is the product of a DFT and adiagonal matrix.

Each set of linearly transformed symbols is then subjected to arespective phase rotation, with the ith set undergoing a phase rotationof θ_(i). The output 22 of the first phase rotation is referred to asA₁, . . . , A_(T), the output 24 of the second phase rotation isreferred to as B₁, . . . , B_(T), and so on, with the output 26 of thePth phase rotation being referred to as P₁, . . . , P_(T).

The rotated sets of symbols 22, 24, . . . , 26 are then input to athreader function 28 which performs a threading operation on the P setsof T phase rotated symbols 22, 24, . . . , 26. The number of threadsproduced is equal to the number of linear transformations P, where2<=P<=M. Each thread consists of an allocation of each one of the set ofT phase rotated symbols to a respective one of the M transmit antennasduring a particular symbol period within an overall codeword durationthat is T symbol period in length according to the two propertiesdefined below. The threaded sequences in turn define M transmitsequences, namely that which will be transmitted by each antenna.

A threaded sequence is an allocation of space-time resources (i.e.antennas over time) of P phase rotated vectors of length T. At eachtime, a thread uses only one antenna, and over time all the antennas arepreferably used equally to the extent possible. If the number ofantennas is not an even multiple of length T, then it will not bepossible to use all the antennas equally in each thread. At the veryleast, each antenna is used at least once for each thread.Mathematically, a thread can be thought of as a placement of ‘1’s in apermutation matrix.The rate R of the system is (P×K transmitted symbols)/(T symbolperiods)=number of symbols transmitted per transmit symbol period.

A first simple example of threading is shown in FIG. 1 where the outputof the linear transformations is a 3×3 matrix generally indicated at 30and the output of the threader is generally indicated at 32. In thisexample, it is assumed the input to each linear transformation is also 3symbols so that P=M=K=T=3. In this example, the three phase rotatedsequences {A₁, A₂, A₃}, {B₁, B₂, B₃}, {C₁, C₂, C₃} are transformed intothreaded sequences {A₁(Tx1), A₂(Tx2), A₃(Tx3)} (this thread indicated at34 in the Figure), {B₁(Tx2), B₂(Tx3), B₃(Tx1)}, and {C₁(Tx3), C₂(Tx1),C₃(Tx2)}. The code rate R for this code is 3×3/3=3. The threadedsequences define the transmit sequences that are transmitted by eachantenna. In particular it can be seen that the transmit sequence {A₁,C₂, B₃} is transmitted from the first antenna Tx1, the transmit sequence{B₁, A₂, C₃} is transmitted from the second antenna Tx2, the transmitsequence {C₁, B₂, A₃} is transmitted from the third antenna Tx3.

A second simple example of threading is shown in FIG. 1 where the outputof the linear transformations is a 3×2 matrix generally indicated at 40and the output of the threader is generally indicated at 42. In thisexample, the two phase rotated sequences {A₁, A₂, A₃}, {B₁, B₂, B₃} aretransformed into two threaded sequences {A₁(Tx1), A₂(Tx2), A₃(Tx3)},{B₁(Tx2), B₂(Tx3), B₃(Tx1)}. In can be seen that {A₁, 0, B₃} istransmitted from the first antenna Tx1, {B₁, A₂, 0} is transmitted fromthe second antenna Tx2, and {0, B₂, A₃} is transmitted from the thirdantenna Tx3. In this example, P=number of threads=2; T=number of symbolsoutput by each linear transformation=3; K=number of symbols input toeach linear transformation is some number <=3, assume it is 2; M=numberof antennas=3. The code rate R for this code is 2×3/3=2. This can beseen from the form of the matrix 36 which shows two symbols beingtransmitted per symbol period.

The structure of FIG. 1 is preferably implemented in a modem thatconverts the output of the channel coder into M antenna signals. Thiscan be implemented in hardware, software, firmware, or a suitablecombination of these functions. It is also to be clearly understood thatwhile a series of functional blocks are shown in FIG. 1 interconnectedin a particular way, the same functionality may be deliverable in alarge number of physical implementations, and moreover, anyimplementation which performs operations which are mathematicallyequivalent to those of FIG. 1 is also considered to fall within thescope of the invention.

An embodiment of the invention provides an M antenna transmitter adaptedto implement one of the linear threaded algebraic space-timeconstellations described herein. The full diversity algebraicconstellations in other embodiments are used in a single antennatransmitter, including but not restricted to, orthogonal frequencydivision multiplexing. An example of this is shown in FIG. 3 where theoutputs of threader 28 are shown input to an IDFT (inverse discreteFourier transform) 50. The output of the IDFT is passed through parallelto serial converter 52 and then to single output antenna 54.

In another embodiment, the threaded algebraic space time constellationdescribed herein may be used in collaborative coding schemes for ad hocwireless networks. A collaborative scheme might be advantageous in thecontext of mobile nodes that do not have sufficient antennas. A mobilenode can use other mobile nodes in its neighbourhood as “virtual”antennas to relay its message to the destination. Assuming that each ofthese nodes is capable of receiving and decoding a source message, thenit can encode it and send it over time. For example, with reference tothe specific example of FIG. 1, a three different nodes could transmit{A₁, C₂, B₃}, {B₁, A₂, C₃} and {C₁, B₂, A₃} respectively.

First Example

Further details of this example can be found in M. O. Damen, H. El Gamaland N. C. Beaulieu, “Linear Threaded Space-Time Constellations,” IEEETransactions on Information Theory, vol. 49, pp. 2372-2388, October2003, hereby incorporated by reference in its entirety.

I. Notation

The following notations are used throughout this example: The letter Rdenotes a ring, Z the ring of integers, Z[i] the ring of complexintegers (or Gaussian integers with iΔ√{square root over (−1)}), Z[j]the ring of Eisenstein integers (with jΔe^(2iπ/3)). The letter F denotesa field, Q the field of rational numbers, R the field of real numbers,and C the field of complex numbers. For n integer, w_(n) Δe^(2iπ/n)denotes the n-th root of unity, and Z[w_(n)] is the ring of algebraicintegers in the n-th cyclotomic number field Q(w_(n)). For m and nintegers, (n|m) denotes their greatest common divisor (if m and n areco-prime, then (n|m)=1). The number of integers less than n and co-primewith it is given by the Euler Φ-function Φ(n). The letter S denotes amultidimensional constellation carved from R^(m) where the ring R andthe dimension m are determined from the context. Further, uppercaseboldface letters denote matrices and lowercase boldface letters denotevectors, with the symbols I and 0 denoting, respectively, the identitymatrix and the all-zero column vector of a size determined by thecontext. Finally, the superscripts ^(T), ^(H), and *, denote thetranspose, hermitian, and conjugate operators, respectively.

The following definitions are useful for the development of our results.

Definition 1. A number ring R⊂C is the ring of algebraic integers in anumber field (i.e., a finite degree extension of the rational numbersfield Q). The shortest non-zero element in R is denoted by Ω_(R), i.e.,Ω_(R) Δinf{|s|,s≠0εR} (note that Ω_(R) is not necessarily attained by anonzero element from R). Finally, we denote by F the field of fractionsof R, i.e., FΔ{α/β:α,βεR,β≠0} (e.g., R=Z, F=Q). In the sequel werestrict our attention to multidimensional constellations over numberrings S⊂R^(m), where a special attention is devoted to cyclotomic numberrings.Definition 2. A unitary transformation UεC^(mxm), with UU^(H)=I, iscalled fully diverse over the ring R if its product distance is nonzeroover UR^(m)\{0}. In particular, let sΔ(s₁, . . . s_(m))^(T) ≠OεR^(m),and xΔ(x₁, . . . , x_(m))^(T)=Us, then U is fully diverse if

$\begin{matrix}{{{{D_{U}(s)}\underset{\underset{\_}{\_}}{\Delta}\tau{\prod\limits_{k = 1}^{m}{x_{k}}}} \neq 0},\mspace{14mu}{\forall{{s \neq O} \in R^{m}}}} & (1)\end{matrix}$where τΔm^(m/2) is a normalization factor introduced here only for thesake of simplification (as evident in the proof of Lemma 3). If U isfully diverse then all constellations carved from UR^(m) have the fulldiversity property. We defined the minimum product distance of U over amultidimensional constellation S⊂R^(m) as

$\begin{matrix}{{d_{U}(S)}\underset{\underset{\_}{\_}}{\Delta}\frac{\min}{s \in {s{{\backslash(} o )}}}{{D_{U}(s)}.}} & (2)\end{matrix}$We call U optimal over S if for any unitary transformation V, we haved_(U)(S)≧d_(V)(S). We call U optimal over R^(m) if ∃S₀⊂R^(m) such thatfor any unitary transformation V, we haved _(u)(S)≧d _(v)(S),∀S⊂R ^(m) such that S

S ₀.  (3)

The optimality criterion adopted here is inspired by the minimum productdistance of unconstrained multidimensional constellations carved fromR^(m) with mεZ⁺².

II. The Construction

Two lemmas that characterize optimal unitary transformations over numberrings and follows:

Lemma 3. Let U be a unitary transformation. When the elements of U,u_(kl), k,l=1, . . . , m, belong to a field K (or a ring R′) thatcontains the number ring R, then U is optimal over R^(m) ifD_(U)(S)εR\{0},∀s≠0εR^(m).

Proof. Let s=(S₁, 0, . . . , 0)^(T), then

$\begin{matrix}{{D_{U}(s)} = {\tau{s_{1}}^{m}{\prod\limits_{k = 1}^{m}{{u_{k\; 1}}.}}}} & (4)\end{matrix}$Since UU^(H)=I, we have

$\begin{matrix}{{\sum\limits_{k = 1}^{m}{u_{k\; 1}}^{2}} = 1.} & (5)\end{matrix}$Applying the Cauchy inequality theorem, we obtain

$\begin{matrix}{{( {\prod\limits_{k = 1}^{m}{u_{k\; 1}}^{2}} )^{\frac{1}{m}} \leq \frac{\sum\limits_{k = 1}^{m}{u_{k\; 1}}^{2}}{m}} = {\frac{1}{m}.}} & (6)\end{matrix}$Substituting the above inequality in (4), we conclude

$\begin{matrix}{D_{U} = {{\tau{s_{1}}^{m}{\prod\limits_{k = 1}^{m}{u_{k\; 1}}}} \leq {{s_{1}}^{m}.}}} & (7)\end{matrix}$It follows that ∃S₀ such that ∀S

S₀d _(U)(S)≦min{|s| ^(m) εS _(2D) ,s≠0}≦min{|s|εS _(2D) ,s≠0},where S_(2D) is the range (in R) spanned by the components of S and thelast inequality follows since R contains 1. Therefore, as theconstellation S grows, the best minimum product distance one can attainapproaches Ω_(R) as S tends to R^(m). Hence, U is optimal ifD_(U)(s)εR\{0},∀s≠0εR^(m).

We note here that full diversity is a necessary condition foroptimality. This lemma specifies the notion of optimality of thetransformation U, especially when one has a lower bound on the minimalabsolute value of the elements of R (e.g., for R=Z,Z[i],Z[j], one hasΩ_(R)=1). In fact, when Ω_(R)=1, the optimal unitary transformation willmaximize the minimum product distance for all finite constellationscarved from R. Since all known digital modulations are carved from thecyclotomic integer rings Z[w_(n)], it is important to know the optimalvalues of the minimum product distances as a function of n. One canprove the following:

Lemma 4.Ω_(z[w) _(n) _(])=1, n=1,2,3,4,6  (8)Ω_(z[w) _(n) _(])=0, n=5 or n>6  (9)

This result is directly related to the fact that the constellationscarved from Z, Z[i], and Z[j] have a constant minimum squared Euclideandistance of 1 (within a normalization depending only on the energy ofthe constellation). In these cases, the optimal unitary transformationswill achieve the optimal minimum product distance for any finiteconstellations.

The PSK constellations carved from Z[w_(n)] (with n=5, or n>6), on theother hand, have a minimum Euclidean distance that goes to zero as nincreases. The notion of full diversity still holds in these cases aswell since d_(U)(S) tends to zero when increasing the size of theconstellation, but can be guaranteed to never attain this value for anyfinite constellation S. As shown in the following, our construction willguarantee a non-zero lower bound on the minimum product distance thatdepends on the size of the constellation.

We observe that, in practice, it may not be beneficial to constructmulti-dimensional constellations using PSK inputs since, by doing so,one looses the constant modulus property of these constellations.Nonetheless, proposing good unitary transformations over Z[w_(n)] isinteresting for other constellations carved from this ring which havegood minimal squared Euclidean distances that do not vanish in thelimit. Such constellations can arise from rotating m-dimensionalconstellations over Z or Z[i] by unitary transformations.

Theorem: Let R be a number ring, φ₁, . . . , φ_(m) with |φ_(k)|=1, . . ., m, and F_(m) be the m×m Discrete Fourier Transform (DFT) matrix withentries

$\begin{matrix}{{f_{kl} = {{\frac{1}{\sqrt{m}}{\mathbb{e}}^{{- 2}{{\mathbb{i}\pi}{({l - 1})}}{{({k - 1})}/m}}} = {\frac{1}{\sqrt{m}}( w_{m}^{{({k - 1})}{({l - 1})}} )^{\star}}}},l,{k = 1},\ldots\mspace{14mu},{m.}} & \;\end{matrix}$Then, the unitary transformationUΔF_(m) ^(H)diag(φ₁,φ₂, . . . , φ_(m))  (10a)is fully diverse over R if φ₁=1, φ₂=φ^(1/m), . . . , φ_(m)=φ^((m−1)/m),and φ is chosen such that {1, φ, . . . , φ^(m−1)} are algebraicallyindependent over R. Such a choice of φ includes:

-   -   1. φ transcendental: φ=e^(iλ), with λ=0εR algebraic.    -   2. φ algebraic of degree≧m over F, the field of fractions of R,        such that {1, φ, . . . , φ^(m−1)} is a basis or part of a basis        of Q(φ) over F (i.e., the degree of φ over F is m′≧m such that        {1, φ, . . . , φ^(m′−1)} is a basis of Q(φ) over F).        More strongly, when R=Z[w_(n)], our construction yields optimal        unitary transformations if and only if        φ=w _(n), when n is even: n=2^(t) ⁰ ×p ₁ ^(t) ¹ × . . . ×p _(f)        ^(t) ^(f)   (10b)        φ=w _(2n), when n is odd: n=p ₁ ^(t) ¹ × . . . ×p _(f) ^(t) ^(f)          (10c)        m=2^(r) ⁰ ×p ₁ ^(r) ¹ × . . . ×p _(f) ^(r) ^(f)   (10d)        where P₁, . . . , P_(f) are primes different from 2, and f, t₀,        . . . t_(f), r₀ . . . , r_(f), are positive integers. In        particular, R=Z[i] implies that the optimal unitary        transformations using our construction in (10) are only possible        for m=2^(r), r≧0 and R=Z[j] implies that they only exist for        m=2^(r) ⁰ 3^(r) ¹ , r₀≧0, r₁≧0.

If one restricts R=Z[w_(n)], then full diversity unitary transformationscan still be found for m not satisfying (4) with good minimum productdistances. Achieving the optimal minimum product distance using ourconstruction is, however, not possible in these scenarios since φ shouldbe chosen of a higher degree than the degree of w_(n) in order to maked_(U)(S)≠0 for S⊂Z[w_(n)]^(m). From the proof of the Theorem, one cansee that this implies d_(U)(s)∉Z[w_(n)] for arbitrary SεZ[W_(n)]^(m);indeed, its value expresses the simultaneous Diophantine approximationof {1, φ, . . . , φ^(m−1)} by numbers from Z[w_(n)]. Therefore, choosingφ such that {1, φ, . . . , φ^(m−1)} are badly approximated over Z[w_(n)]enhances the minimum product distance. One possible choice of φ is analgebraic integer with the smallest degree that satisfies the Theorem.In this case, by choosing φ to be algebraic such that {1, φ, . . . ,φ^(m−1)} is a basis or part of a basis of Q(φ) over Z[w_(n)], one hasthe following lower bound on minimum product distance of a finiteconstellation S⊂Z[w_(n)]^(m)

$\begin{matrix}{{{d_{U}(S)} \geq \frac{1}{( {2 + {mv}} )^{m{({d - 1})}}}},} & (11)\end{matrix}$where d is the minimal degree of the number field that contains φ andthe entries of S (i.e., d is the maximum of the degrees of φ and w_(n)),and v is a quantity that depends only on the maximum absolute value ofthe components of the vectors in the finite constellation S. The proofis directly related to the simultaneous approximation of algebraicnumbers by other algebraic numbers. In order to maximize this lowerbound (5), one needs to choose φ algebraic with the smallest possibledegree which satisfies the main theorem. For example, for m=5, and n=4,one needs to choose φ=w_(q) of degree Φ(q)≧m. One can readily verifythat the smallest number satisfying this is q=7. Indeed, we have foundthat φ=w₇ gives a local optimal value, with respect to the possiblechoices of φ, for the minimum product distance when using a 4-QAMconstellation carved from Z[i]. Although d_(U)(S) may decrease whenincreasing the size of the constellation in this case, one can easilysee from our result that this choice of φ gives a fully diversetransformation over all constellations carved from Z[i]. As a generalrule to maximize the lower bound in (5) for n=4 (when m is not a powerof 2), one should choose φ=w_(q) with q the first prime such that q−1≧m.This rule also applies for n and m not satisfying (4) where one choosesφ=w_(q) with q the smallest prime such that {1, φ, . . . , φ^(m−1)} areindependent over Z[w_(n)]. This rule, however, does not always give the(global) optimal φ for a given finite constellation carved fromZ[w_(n)]. In Table I we present the minimum product distances achievedby the proposed unitary transformation with a normalized 4-QAM input fordifferent values of m. For m=5, 6, 7, φ chosen by the above rule isfound to give local optima of the minimum product distances; however,this is not the case when choosing φ=w₅ for m=3. In this special case,we have found φ=w₃ and φ=w₃₅ ³ to give local optima of the minimumproduct distance over this constellation of 0.268 and 0.4854,respectively. As expected, for m satisfying (4), φ=w₄ gives globaloptima. In Table II we report the minimum product distances of theproposed rotations with m=2 and n-PSK input constellations (with n=4, 8,16, 32, 64). As predicted by the optimality criterion of Lemma 3, wefound φ=w_(n) to yield global optima in all these cases. Also, note thatthe exponent in the lower bound in (5), m(d−1), is minimized when φ andm satisfy the constraints (11)-(4), i.e., when the optimality criterionin Lemma 3 is satisfied.Remarks:1) The general construction in (10) allows for more degrees of freedomby simultaneously optimizing φ₁, . . . , φ_(m), especially for thedimensions m that are not powers of the primes dividing n This line ofresearch is under investigation. The transcendental choice of φ can beof interest in some cases. For example, selecting φ randomly over theunit circle gives a fully diverse unitary transformation withprobability 1. This is because the probability that φ will be algebraicis zero since the algebraic numbers are of measure zero in C.

TABLE I The Minimum Product Distances When Using Construction (10a) OverDifferent Rotated m-Dimensional 4-QAM Constellation ⊂ Z[i] m 2 3 4 5 6 78 d_(U) (s) 1 0.0403 1 0.0543 0.0568 0.0024 1

TABLE II The Minimum Product Distances When Using Construction (10a)Over Different Rotated 2-Dimensional n-PSK Constellations ⊂ Z[w_(n)] n 48 16 32 64 d_(U) (s) 1 0.2241 0.0297 0.0037 4.7255e−04

TABLE III The Minimum Product Distances When Using Constellation (10a)Over Different Rotated m-Dimensional 8-QAM Constellations ⊂Z[i]∪Z[j] m 23 4 d_(U)(s) 0.2188 0.0287 0.02482) Instead of choosing φ₁, . . . , φ_(m) on the unit circle, one canallow them to be chosen from C, where U is normalized by the norm of thevector (φ₁, . . . , φ_(m)) to ensure constant average power. This can beuseful for example if one wishes to use full diversity realconstellations, or to allocate different powers to different symbols asin signal space coding for multi-user applications. For example, whenm=2, the product distance is given byD _(U)(S)=|φ₁ ² s ₁ ²−φ₂ ² s ₂ ²|  (12)with SεZ². It is clear that choosing φ₁ and φ₂ such that

$\frac{\phi_{1}^{2}}{\phi_{2}^{2}}$is not a square in Q guarantees the full diversity property. Asdiscussed in Lemma 3, if one can further guarantee that D_(U)(s)εZ⁺ forall s≠(0,0)^(T) then one optimizes the minimum product distance. Suchchoice includes φ₁=1 and φ₂ any quadratic residue (√{square root over(2)}, √{square root over (3)}, . . . ).3) Some important digital modulations may belong to a union ofcyclotomic number rings, e.g., the most energy efficient 8-QAMconstellation given by {1+i, 1−i, −1+i, −1−i, α, −α,αi,−αi}⊂Z0[i]ÅZ0[j], with α=1+√{square root over (3)}. It isstraightforward to apply our construction to this case using the rulesabove. For example, Table III lists the minimum product distances of theproposed rotated m-dimensional constellations with inputs from the above(normalized) 8-QAM constellation. Local optima are obtained for φ=w₄w₆,w₅w₆, and w₄w₆, respectively.Method of Constellation Construction

Another embodiment provides a method of constructing a full diversityalgebraic constellation that might for example be used with thestructure of FIG. 1. The method will now be described with reference toFIG. 2. The method starts with the selection at step 2-1 of aparameterized linear transformation. The linear transformation isparameterized in the sense that it defines a class of transformations asa function of at least one parameter over which optimization can beperformed. Examples of suitable parameterized linear transformationsinclude the previously introduced transformation of equation 10a. Inother embodiments, other transformations are employed, but preferably afull diversity transformation is employed.

The next step 2-2 involves performing an optimization of the lineartransformation. An optimization is achieved by determining the bestparameter/parameters of the linear transformation as defined by anobjective function. In a preferred embodiment, the objective function isone which maximizes coding gain of the linear transformation.

In the example of equation 10a, the parameters include φ₁, φ₂, . . . ,φ_(M) used in the diagonal matrix diag (φ₁, φ₂, . . . , φ_(M)).Preferably, these are defined as a function of a single parameter φ.Optimization is performed to determine the value of φ which results inthe best coding gain for the linear transformation. Closed form resultsare available in certain circumstances, and these results are given inequation 10b, 10c and 10d for some codes. A true optimal result can onlybe determined for certain cases. Full diversity unitary transformationscan still be found for m not satisfying 10d with good, but notnecessarily optimal, minimum product distances. In this case, thisoptimization step is better referred to as a near-optimization step.

The next step 2-3 involves performing a further optimization todetermine the phase rotations φ_(i), for i=0, . . . , M−1. This isachieved by determining the best set of phase rotations as defined by anobjective function. In a preferred embodiment, φ_(i)=φ^(i−1) for somevalue φ. In such embodiments, the further optimization need onlydetermine the single optimal value for φ. The following is an example ofa solution for φ.

In a preferred embodiment, in which it is assumed that K=M, codes arecompletely specified by the parameters M (the number of transmitantennas), N (the number of receive antennas), T (the number of symbolperiods, i.e. the code consists of M×T signalling matrices), P (thenumber of threads giving the rate of the code in symbols from the inputalphabet per channel use; we set P=min (M,N) in order to achieve fullrate), θ (the Diophantine number within each thread in the M×Tspace-time matrix) and φ (the Diophantine number between the threads).The methods described above allow for a systematic design of space-timecodes that achieve full diversity for arbitrary M, T, P and inputalphabets. Given below in Table 1 is a set of examples of the new codesfor some values of the above parameters with QAM constellations as inputalphabets. Note that the values of θ and φ have been optimized forenhanced coding gain.

TABLE 1 M N T P θ φ 2 1 2 1 exp(iπ/4) 1 3 1 3 1 exp(2π/5) 1 4 1 4 1exp(iπ/8) 1 2 2 2 2 exp(iπ/4) exp(iπ/6) 4 2 4 2 exp(iπ/8) exp(iπ/6) 3 33 3 exp(2π/5) exp(2π/7) 4 4 4 4 exp(iπ/8) exp(iπ/32) 6 6 6 6 exp(2π/7)exp(2π/42) 8 4 8 4 exp(iπ/16) exp(iπ/64) 10 5 10 5 exp(2π/11) exp(2π/55)12 12 12 12 exp(2π/13) exp(2π/156)

Specific examples of codes provided by embodiments of the invention aredefined in equation 27 of the second example below for a MISO channel,and equations 32, 33, 34, 35, 36 of the second example for MIMOchannels.

Two specific performance examples are shown in Table 2 and Table 3below. In both of these examples, the row “new code” contains the BERperformance of the new code defined in equation 34 of the secondexample. The row “TI code” contains the BER performance for space-timeblock coded transmit antenna diversity for W-CDMA, defined in SMG2document 581/98, submitted October 1998. The row “Lucent code” containsthe BER performance results for the BLAST code developed by Lucent.

The results in Table 2 are for a rate of 6 bits/Hz, with two receiverantennas. The results in Table 3 are for a rate of 8 bits/Hz. It can beseen that for both examples, and at various SNRS, the bit error rate ofthe new code is superior to the two other codes, in some casessignificantly so.

TABLE 2 Rate 6 bits/s/Hz (Two receiver antennas) SNR 10 dB 15 dB 20 dB25 dB New Code 1.074943e−01 2.702250e−02 2.915833e−03 1.671667e−04 TICode 1.177350e−01 3.278803e−02 4.496805e−03 1.783414e−04 Lucent Code1.080808e−01 2.798733e−02 3.602167e−03 2.930000e−04

TABLE 3 Rate 8 bits/s/Hz (Two receiver antennas) SNR 20 dB 25 dB 30 dB35 dB New Code 1.214286e−02 1.042427e−03 5.352218e−05 1.044500e−06 TICode 3.771820e−02 5.649718e−03 3.073882e−04 3.338362e−05 Lucent Code1.227310e−02 1.267596e−03 1.338178e−04 6.216917e−06

The space-time coder alone may be of value in a collaborative scheme inan ad hoc wireless network. The invention is not limited to wirelessapplications but may find application in data storage and retrievalsystems for audio, video data, and more.

Second Example

Further details of this example can be found in M. O. Damen, H. El Gamaland N. C. Beaulieu, “Systematic Construction of Full Diversity AlgebraicConstellations,” IEEE Transactions on Information Theory, vol. 49, pp.3344-3349, December 2003, hereby incorporated by reference in itsentirety. For this example, equation numbering is restarted at (1).

I. System Model

We consider signaling over an M×N multi-input multi-output (MIMO)channel. A K×1 information symbol vector u=(u₁, . . . , u_(K))^(T)εU^(K), where U denotes the input constellation, is mapped by aconstellation encoder γ into an MT×1 output vector γ(u), with componentsfrom the alphabet

$\prod\limits_{i = 1}^{MT}S_{i}$(i.e.,

$ {\gamma: U^{K}arrow{\prod\limits_{i = 1}^{MT}S_{i}} } ).$In general, we allow the transmitted single dimensional constellationS_(i) to vary across time and space. All the proposed constellations,however, will enjoy the symmetry property that |S_(i)|=|S_(j)|. In thiswork, we also assume that the information symbol vector is a randomvariable with a uniform distribution over U^(K). A space-time formatter,“F”, then maps each encoded symbol vector γ(u) into an M×T space-timeconstellation C_(u)=F(γ(u)), where M symbols C_(mt) (m=1, . . . , M) aretransmitted simultaneously from the M transmit antennas at time t, t=1,. . . , T. When there is no confusion, we denote the space-timeconstellation by C. The transmission rate of the constellation C istherefore equal to K/T symbols per channel use (PCU). The throughput ofthe system, in bits PCU, is therefore given by K/T log₂|U|.

The N×T received signal matrix X, after matched filtering and samplingat the symbol rate, can be written asX=√{square root over (ρ)}HC _(u) +W,  (1)where H=[h_(nm)] is the N×M channel matrix, and h_(nm) denotes thefading coefficient between the m^(th) transmit and the n^(th) receiveantenna. These fading coefficients are assumed to be independent,identically distributed (i.i.d.) zero-mean complex Gaussian randomvariables with unit variance per complex dimension. In the quasi-static,frequency non-selective fading model adopted in this paper, the fadingcoefficients are assumed to be fixed during one codeword (i.e., T timeperiods) and change independently from one codeword to the next. Theentries of the N×T noise matrix W, i.e., w_(nt), are assumed to beindependent samples of a zero-mean complex Gaussian random process withunit variance per complex dimension. We further impose the average powerconstraint that

$\begin{matrix}{{\sum\limits_{m = 1}^{M}{\sum\limits_{t = 1}^{T}{E_{u}{c_{mt}}^{2}}}} = T} & (2)\end{matrix}$where E_(u) refers to expectation with respect to the random data vectoru. The received SNR at every antenna is therefore independent of thenumber of transmit antennas and is equal to ρ. Moreover, we assume thatthe channel state information (CSI) is available a-priori only at thereceiver. Unless otherwise stated, we focus our attention onconstellations U⊂ξZ[w_(M)], where ξ is a normalization constant and

$\begin{matrix}{{Z\lbrack w_{n} \rbrack}\overset{\Delta}{=}\{ {{\sum\limits_{k = 0}^{{\phi{(M)}} - 1}\;{a_{k}w_{M}^{k}}},{a_{k} \in Z},{k = 0},\ldots\mspace{14mu},{{\phi(M)} - 1}} \}} & (3)\end{matrix}$is the ring of integers of the M-th cyclotomic number field, Q(w_(M)),with Z the ring of integer numbers, w_(M)=e^(2iπ/M) the M-th primitiveroot of unity, and φ(M) denoting the Euler φ-function that measures thenumber of integers less than M and co-prime with it. Without loss ofgeneralization, we will assume that ξ is adjusted to normalize theaverage power of U to one (i.e., E[|u|²]=1). With a slight abuse ofnotation, we will rely on the isomorphism between ξZ[w_(M)] and Z[w_(M)]and refer to both rings as Z[w_(M)] when there is no confusion. We alsodenote the minimum squared Euclidean distance of U as d_(U) ². Finally,we note that this set of constellations contains all the pulse amplitudemodulation (PAM) constellations (i.e., M=1, 2), square quadratureamplitude modulation (QAM) constellations (i.e., M=4), constellationscarved from the hexagonal lattice (i.e., M=3), and phase shift keying(PSK) constellations (i.e., M≧5).

By stacking all the columns of matrix X in one column, i.e., xΔvec(X),the received signal in (1) can be written in a vector form asx=√{square root over (ρ)}Hc _(u) +w,  (4)where HΔI_(T)

HεC^(TN×TM),

denotes the Kronecker matrix product, c_(u) Δvec(C), and wΔvec(W), withC the field of complex numbers. If the alphabet U belongs to a numberring R (e.g., U=4-QAM ⊂R=Z[i]), then one calls the constellation Clinear over R if C_(u′)+C_(u″)=C_(u′+u″), for u′,u″εU^(K). In this casethere exists a generator matrix MεC^(TM×K) such that c_(u)=Mu. Then, (4)is a linear system with N×T equations and K unknowns with the combiningmatrix HM. The maximum likelihood (ML) solution in this scenario can beimplemented using a sphere decoder whose average complexity is onlypolynomial in K for K≧N×T and medium to large SNR's. For K>N×T, one canuse the generalized sphere decoder whose complexity is exponential inK−N×T and polynomial in N×T.

The main reason for restricting the discussion in this paper to linearconstellations is to benefit from the linear complexity space-timeencoder and the polynomial complexity ML decoding allowed by thelinearity property when K≦N×T. The linearity of the constellation,however, implies some fundamental performance limits as detailed in thefollowing two sections.

II. Design Criteria and Fundamental Limits

One of the fundamental challenges in the design of space-time signals isthe fact that the optimal design criteria depend largely on the systemparameters (e.g., number of receive antennas) and quality of serviceconstraints (e.g., maximum allowable delay). One of the advantages ofthe proposed TAST constellations is that they nearly optimize thesedifferent criteria simultaneously.

1) Diversity Order and Coding Gain: Under the quasi-static assumption,the Chernoff upper bound on the pairwise error probability of the MLdetection of u″ given that u′≠u″ was transmitted is given by

$\begin{matrix}{{{\Pr\{ u^{\prime}\longrightarrow u^{''} \}} \leq ( {\det( {I_{M} + {\frac{\rho}{4}{A( {u^{\prime},u^{''}} )}}} )} )^{- N}} = ( {\prod\limits_{l = 1}^{r{({A{({u^{\prime},u^{''}})}})}}( {1 + {\frac{\rho}{4}{\lambda_{l}( {u^{\prime},u^{''}} )}}} )} )^{- N}} & (5)\end{matrix}$where I_(M) is the M×M identity matrixA(u′,u″)Δ(C_(u′)−C_(u″))(C_(u′)−C_(u″))^(H)the superscript ^(H) denotes the conjugate transpose operator,r(A(u′,u″)) is the rank of A(u′,u″), and λ₁(u′,u″), . . . ,λ_(r(A(u′,u″)))(u′,u″) are the non-zero eigenvalues of A(u′,u″). One caneasily see that the largest power of the inverse of the SNR (i.e.,diversity order) in (5) is equal to N×r(A(u′,u″)), and the dominant termin (5) at large SNR is

$\prod\limits_{l = 1}^{r{({A{({u^{\prime},u^{''}})}})}}{{\lambda_{l}( {u^{\prime},u^{''}} )}.}$

This observation gives rise to the well known determinant and rankcriteria. Therefore, a full diversity space-time constellation Cachieves the maximum diversity order of MN. In addition, one refers tothe term

$\begin{matrix}{\delta_{C}\overset{\Delta}{=}{\frac{\min}{{u^{\prime} \neq u^{''}} \in U^{K}}( {\prod\limits_{l = 1}^{M}{\lambda_{l}( {u^{\prime},u^{''}} )}} )^{1/M}}} & (6)\end{matrix}$as the coding gain of the full diversity constellation C.

Using the linearity of the constellation, the average power constraint,and the geometric mean/arithmetic mean inequality, one can see that themaximum achievable coding gain for a linear space-time constellationthat supports L symbols PCU from an input constellation U is given by

$\begin{matrix}{\delta_{C} \leq {\frac{d_{U}^{2}}{ML}.}} & (7)\end{matrix}$2) Squared Euclidean Distance: For small SNR and/or large numbers ofreceive antennas, one can see that the dominant term in (5) is thesquared Euclidean distance of space-time constellation C given by

$\begin{matrix}{d_{C}^{2}\overset{\Delta}{=}{\frac{\min}{u^{\prime},{u^{''} \in U^{K}}}{\sum\limits_{l = 1}^{M}{{\lambda_{l}( {u^{\prime},u^{''}} )}.}}}} & (8)\end{matrix}$Again, using the linearity and average power constraints, one can showthat

$\begin{matrix}{{d_{C}^{2} \leq \frac{d_{U}^{2}}{L}},} & (9)\end{matrix}$for a linear space time constellation that supports L symbols PCU.

One can use (7) and (9) to extract a useful design guideline. Thenon-linear shrinking of d_(U) ² with the size of the constellationimplies that the upper bounds in (7) and (9) are maximized by maximizingthe number of symbols PCU for a fixed throughput. The only exception forthis rule is when one moves from a BPSK to a QPSK constellation wherethe two choices are equivalent. This exception can be attributed to thewasteful nature of the BPSK constellation. The maximum value of L is,however, limited to min(N,M) to facilitate polynomial complexity MLdecoding. This argument implies that the choice L=min(N,M) strikes avery favorable tradeoff between performance and complexity. Therefore,all the proposed constellations will be constructed to achieve min(N,M)symbols PCU. Moreover, we will show that the proposed constellationsachieve the upper bound on the squared Euclidean distance with equalityin all cases.

3) Peak-to-Average Power Ratio (PAR): The PAR of the space-timeconstellation plays an important role in peak power limited systemsbecause a high value of the PAR will shift the operating point to thenonlinear region of the power amplifier which may cause power clippingand/or distortion. Therefore, it is desirable to construct space-timeconstellations with low PAR values. We define the baseband PAR for agiven constellation U as

$\begin{matrix}{{{PAR}_{U}\overset{\Delta}{=}\frac{\max{u}^{2}}{E\lbrack {u}^{2} \rbrack}},{u \in {U.}}} & (10)\end{matrix}$

For example, the PAR for a square M-QAM constellation equals

$3{\frac{\sqrt{M} - 1}{\sqrt{M} + 1}.}$The PAR of the space-time constellation C is given by

$\begin{matrix}{{{PAR}_{C}\overset{\Delta}{=}\frac{\max{c_{mt}}^{2}}{E\lbrack {c_{mt}}^{2} \rbrack}},\mspace{20mu}{c_{mt} \in S_{i}},{i = 1},\ldots\mspace{20mu},{MT}} & (11)\end{matrix}$where S_(i) is the single dimension alphabet at the output of theencoder. For symmetric ST constellations, with the same average powertransmitted from all the antennas and the same PAR for all the S_(i),the average power constraint can be used to simplify (11) toPAR_(C) ΔM max |C₁₁|²,C₁₁εS₁  (12)The linearity of the space-time encoder and the independence of theinputs imply the following lower bound on the PAR of the constellationPAR_(C)≧PAR_(U)  (13)where U is the input constellation. Guided by the single antennascenario, one can see that there is a fundamental tradeoff betweenoptimizing the performance of the constellation in average power limitedsystems and minimizing the PAR. For example, it is well-known that QAMconstellations outperform PSK constellations in terms of average powerperformance while PSK constellations enjoy the optimum PAR. To quantifyand utilize this tradeoff, we define the normalized coding gain andsquared Euclidean distance, respectively, as

$\begin{matrix}{\eta_{C}\overset{\Delta}{=}\frac{\delta_{C}}{{PAR}_{C}}} & (14) \\{X_{C}^{2}\overset{\Delta}{=}\frac{d_{C}^{2}}{{PAR}_{C}}} & (15)\end{matrix}$

These metrics play the same role as the coding gain and squaredEuclidean distance, defined earlier, in the case of peak power limitedsystems. They will be used in the sequel to guide the design and measurethe optimality of the proposed space-time constellations. Combining (7),(9), (10), and (13), we obtain the following upper bounds on thenormalized coding gain and squared Euclidean distance

$\begin{matrix}{{\eta_{C} \leq \frac{d_{U}^{2}}{M \times L \times {PAR}_{U}}},} & (16) \\{\chi_{C}^{2} \leq {\frac{d_{U}^{2}}{L \times {PAR}_{U}}.}} & (17)\end{matrix}$We will show later that these bounds are achievable for constellationsthat support one symbol PCU.4) Delay: One can easily see that a non-zero coding gain, and hence,full diversity, can only be achieved if T≧M (i.e., so that A can havefull row rank). Therefore, the space-time constellation C will be calleddelay optimal if T=M. All the constellations considered in this paperare delay optimal by construction, and hence, we will always assume thatT=M unless otherwise stated. The optimality of the delay is alsodesirable from a complexity point of view since it minimizes thedimension of the sphere decoder.5) Mutual Information: Assuming that the ST constellation will beconcatenated with a Gaussian outer codebook, Hassibi and Hochwaldproposed the average mutual information between the input of thespace-time constellation and the received signal as the design metric.They further presented a numerical optimization technique forconstructing constellations with near optimal average mutualinformation. It is straightforward to see that the optimal constellationis the one that preserves the capacity of the channel, and hence, wewill refer to it as an information lossless constellation. The primeexample of an information lossless constellation is the identity parserwhich distributes the output symbols of the outer code across the Mtransmit antennas periodically. However, optimizing the mutualinformation only may not be sufficient to guarantee good performance.Furthermore, imposing the constraint that L≦min(N,M) generally entails aloss in the mutual information when N<M (the only known exception forthis observation is the Alamouti scheme with one receive antenna). Inthe sequel, we will show that the average mutual information achieved bythe full diversity constellations proposed here is optimal (i.e.,information lossless) when N≧M and very close to being optimal when N<M.III. The Diversity vs. Rate Tradeoff

In multi-antenna systems, one can increase the transmission rate at theexpense of a certain loss in the diversity advantage. Earlier attemptsto characterize this tradeoff have defined the transmission rate as thenumber of transmitted symbols PCU. The TAST constellations presentedhere offer a constructive proof that one can simultaneously achieve fulldiversity while transmitting at the full rate of min(M,N) symbols PCU.The tradeoff between rate and diversity becomes only necessary if oneimposes further requirements on the system. Three scenarios areconsidered in the following subsections. First in section A below weallow the transmission rate to increase with the SNR. Then, wecharacterize the diversity-vs-rate tradeoff under peak power andcomplexity constraints in sections B and C respectively.

A. Rate Scaling with the Signal-to-Noise Ratio

The transmission rate, in bits PCU, is allowed to grow with the SNR asR=r log₂ρ  (18)where r is defined as the multiplexing gain. The authors furthercharacterize the optimal tradeoff between the achievable diversity gaind, 0≦d≦MN, and the achievable multiplexing gain r, 0≦r≦min(M,N), for anM×N MIMO system asd=(M−r)(N−r)  (19)

This characterization has an elegant interpretation for MIMO systemswith fixed transmission rates.

Proposition 1. Let C(R) be a ST signaling scheme that supports anarbitrary rate R in bits PCU. Then, C achieves the optimaldiversity-vs-multiplexing tradeoff ifP _(e)(C(R),ρ)=αP _(out)(R,ρ),∀R,ρ  (20)where ρ is the SNR, P_(out)(R,ρ) is the outage probability at R and ρ,P_(e)(C(R),ρ) is the probability of error at this particular rate andSNR, and α is an arbitrary constant.

Proposition 1 means that the gap between the performance of the optimaltransmission scheme and the outage probability should be independent ofthe transmission rate and the SNR. Proposition 1 also highlights thefact that this tradeoff characterization does not capture the codinggain of the constellation (i.e., the optimal tradeoff curve is achievedfor any constant α). One can therefore augment this tradeoffcharacterization by requiring that α=1 for the optimal scheme.

In the sequel, we will argue that the proposed constellations achievethe optimal tradeoff between the diversity and rate, for N=1 and N≧M,when concatenated with an outer Gaussian codebook under the ML decodingassumption. We will further present simulation results which indicatethat the proposed constellations achieve the optimal tradeoff curve evenwhen the inputs are drawn from uncoded QAM constellations, where theconstellation size increases with the SNR.

B. Tradeoff Under Peak Power Constraints

In order to simultaneously achieve full diversity and full transmissionrate in an unconstrained system, the TAST constellations induce anexpansion of the output constellations S_(i). In fact, thisconstellation expansion is a characteristic of most space-time signalsthat are linear over the field of complex numbers. The constellationexpansion, however, results in an increase in the peak transmittedpower. In order to avoid the increase of the peak power, one can limitthe output constellations (i.e., S_(i)) to be standard, but possiblydifferent, QAM or PSK constellations. This constraint, however, imposesthe following fundamental limit on the tradeoff between transmissionrate and diversity advantage. This bound is obtained from the Singletonbound and assumes a symmetric ST constellation with |S_(i)|=|S_(j)|

$\begin{matrix}{R \leq {\log_{2}{S_{i}}{( {M - \frac{d}{N} + 1} ).}}} & (21)\end{matrix}$For example, to achieve full diversity (i.e., d=MN), the maximumtransmission rate is one symbol, drawn from S_(i), PCU which correspondsto log₂ |S_(i)| bits PCU irrespective of the number of receive antennas.

By combining L symbols from U to obtain a symbol from S_(i), a linear STconstellation can achieve full diversity only if|S _(i) |≧|U| ^(L) ∀i  (22)as predicted by the Singleton bound (21). All the constellationsproposed here satisfy the lower bound in (22) with equality. Now, byimposing the constraint that |S_(i)|=|U| (i.e., no increase in the PAR),equality in the Singleton bound (21) can be satisfied with linear STconstellations only in the full diversity scenario (i.e., d=MN). Ingeneral, the sub-optimality of linear ST constellations in peak powerlimited ST systems is formalized in the following result.Proposition 2. The diversity advantage of a linear ST constellation thatsupports L symbols PCU using output constellations S_(i) that satisfythe constraint that |U|=|S_(i)|, is governed by

$\begin{matrix}{d \leq {\lfloor \frac{M}{L} \rfloor{N.}}} & (23)\end{matrix}$

Variants of the proposed constellations are provided below that realizethis optimal tradeoff. Interestingly, Proposition 2 argues that theV-BLAST architecture achieves the optimal diversity advantage for fullrate symmetric systems (i.e., L=M=N) with strict peak power constraints(i.e., |S_(i)|=|U|).

C. Tradeoff Under Complexity Constraints

Although ML decoding for the full rate and full diversity linearconstellations only requires polynomial complexity in M min(M,N), thiscomplexity can be prohibitive for systems with large numbers of transmitand receive antennas. This motivates the following question: what arethe achievable diversity-rate pairs (d,L) for a MIMO system under theconstraint that the dimension of the sphere decoder is L_(c)? The answerto this question is given in the following proposition.

Proposition 3. In an M×N MIMO system with 1≦L_(c)≦M min(M,N) complexdimensions in the polynomial complexity sphere decoder and a diversityadvantage 1≦d≦MN, the number of transmitted symbols PCU satisfies

$\begin{matrix}{L \leq {\min( {\frac{L_{c}N}{d},M,N} )}} & (24)\end{matrix}$where we require d/N to be an integer.

Proposition 3 means that, with complexity constraints, the choice ofnumber of symbols PCU implies a tradeoff between the diversity advantageand the squared Euclidean distance (since a large number of symbolsresults in a large squared Euclidean distance as evident in (9)). Onecan use this observation, along with the fact that the squared Euclideandistance is the dominant factor for small SNR and/or large number ofantennas, to conclude that the optimal choice of the number of symbolsPCU depends on the available complexity, number of antennas, and SNR.

Similarly, one can investigate the complexity constrained tradeoff forother receiver architectures. For example, we have the followingconjecture for the nulling and cancellation receiver where the diversityadvantage is upper bounded by the number of excess degrees of freedom inthe system of linear equations.

Conjecture 1. In an M×N MIMO system with 1≦L_(c)≦M min(M,N) complexdimensions in the polynomial complexity nulling and cancellationalgorithm supported by the receiver, one has

$\begin{matrix}{\frac{1}{M} \leq L \leq {\min( {M,N,L_{c}} )}} & (25) \\{d \leq {{{\min( {M,\frac{L_{c}}{L}} )}( {N - L} )} + 1.}} & (26)\end{matrix}$D. Constellations for MISO Channels

The designs proposed belong to the threaded algebraic space-time (TAST)signaling framework. The main idea behind this framework is to assign analgebraic code in each thread that will achieve full diversity in theabsence of the other threads. One should then project the threads intodifferent algebraic subspaces by multiplying each one with a properlychosen scaling factor to ensure that the threads are transparent to eachother. Here, we utilize this framework to construct constellations withoptimal (or near optimal) PAR's, minimum squared Euclidean distances,and coding gains. First, we impose the constraint that the number ofthreads is equal to the number of transmit antennas M, rather than thenumber of symbols PCU; this avoids sending zeros from some of thetransmit antennas when L<M. Second, we replace the rate one algebraicrotations used as component codes in the different threads with simplerepetition codes of length M. The resulting constellation, therefore,still supports one symbol PCU while avoiding the increase of PARincurred by the rotation and the periods of no transmission.

TABLE A Coding Gains For M -PSK Constellations With φ = e^(2iπ/M) For MEven And φ = e^(2iπ/2M) For M Odd M 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16δ_(c) 3 1.8 0 −3.7 −3 −7.8 −6.5 −10.9 −9.3 −13.5 −11.6 −15.6 −13.6 −17.4−15.3 (dB)

Mathematically, over M transmit antennas and M symbol periods one sendsM information symbols, u₁, . . . , u_(M) in a circulant M×M matrix asfollows

$\begin{matrix}{{D_{M,M,1}(u)} = {\frac{1}{\sqrt{M}}\begin{pmatrix}u_{1} & \ldots & {\phi^{\frac{1}{M}}u_{2}} \\{\phi^{\frac{1}{M}}u_{2}} & \ldots & {\phi^{\frac{2}{M}}u_{3}} \\\vdots & \ddots & \vdots \\{\phi^{\frac{M - 1}{M}}u_{M}} & \ldots & u_{1}\end{pmatrix}}} & (27)\end{matrix}$where D_(M,M,1) refers to the new TAST constellation with M transmitantennas, M threads, and one symbol PCU. The Diophantine number φ ischosen to guarantee full diversity and optimize the coding gain asformalized in the following two theorems.Theorem 1. If the Diophantine number φ, with |φ|=1, is chosen such that{1, φ, . . . , φ^(M−1)} are algebraically independent over Q(w_(M)), theM-cyclotomic number fields, then D_(M,M,1)(u) in (27) achieves fulldiversity over all constellations carved from Z[w_(M)]. This can beachieved if φ is chosen such that:

-   -   3. φ=e^(iλ) with λ≠0εR algebraic (φ transcendental).    -   4. φ algebraic such that Q(φ) is an extension of degree ≧M over        Q(w_(M)) with {1, φ, . . . , φ^(M−1)} a basis, or part of a        basis of Q(φ) over Q(w_(M)).

Furthermore, D_(M,M,1)(U) achieves the optimal Euclidean distance ofd_(U) ² and the optimal normalized Euclidean distance of

$\frac{d_{U}^{2}}{{PAR}_{U}}$(the constraint |φ|=1 is imposed to ensure this property) Theorem 2. ForM=2^(r),r≧1, the optimal coding gain, i.e.,

${\delta_{C} = \frac{d_{U}^{2}}{M}},$can be obtained with D_(M,M,1) by choosing the Diophantine number φ=iand constellations carved from Z[i], and for M=2^(r) ⁰ 3^(r) ¹ , r₀,r₁≧0 by choosing φ=e^(2iπ/6) and constellations carved from Z[j].

When M≠2^(r), or M≠2^(r) ⁰ 3^(r) ¹ , r₀, r₁≧0, one can only guaranteelocal optima for the coding gains by using exhaustive computer search orby choosing the Diophantine number as an algebraic integer with thesmallest degree that guarantees full diversity (as in Theorem 1). It isalso interesting to note that the optimal choice of Diophantine numbersin Theorem 2 does not depend on the size of the constellation, andhence, the proposed ST constellations are universal for anyconstellation size in these cases.

Theorems 1 and 2 allow for constructing constant modulus full diversityST constellations with polynomial complexity ML decoding for any numberof transmit antennae, M. This can be achieved by using M-PSK inputmodulations, i.e., U□{e^(2iπk/M), k=0, . . . , M−1}⊂Z[w_(m)] andchoosing the Diophantine number φ to be a root of unity which satisfiesthe constraint that {1, φ, . . . , φ^(M−1)} are algebraicallyindependent over Z[w_(M)]. For a given PSK constellation, one can usealgebraic methods combined with computer search, to find φ thatmaximizes the coding gain of the system considered. For example, for M=2with the M-PSK constellation, one has the following relation for thecoding gain

$\begin{matrix}{{\delta_{D_{2,2,1}}(\phi)}\overset{\Delta}{=}{\frac{1}{M}{\min\limits_{{{{({u_{1},u_{2}})} \neq {({0,0})}} \in {Z{\lbrack w_{M}\rbrack}}^{2}},{{u_{k}} \leq 2},{k = 1},2}{{u_{1}^{2} - {\phi\; u_{2}^{2}}}}}}} & (28)\end{matrix}$where the condition |u_(k)|≦2, k=1, 2, ensures that u_(k) is adifference of two points in the M-PSK constellation. Thus, it sufficesto choose φ not to be a quadratic residue in Z[w_(M)] (i.e.,φ^(1/2)∉Z[w_(M)]) in order to guarantee a nonzero determinant. For evenvalues of M, the only roots of unity in Q(w_(M)) are the M-th roots ofunity, and hence, one can choose φ=e^(2iπ/M) in these cases. For oddvalues of M, the only roots of unity in Q(w_(M)) are the 2M-th roots ofunity, and hence, it suffices to one choose φ=e^(2iπ/2M) to guaranteethat φ is not a quadratic residue in Z[w_(M)] in these cases. This way,we can also guarantee that the determinant value in (31) is a nonzerointeger from Z[w_(M)]. Furthermore, it can be shown these values of φmaximize the coding gain for constant modulus transmission with twotransmit antennas (Table 1 reports the optimized coding gains for M=2, .. . , 16). For an arbitrary number of transmit antennas and arbitraryM-PSK constellations, one can construct full diversity TASTconstellations with optimal PAR's and optimized coding gains by settingthe Diophantine number φ according to the rules in D. A. Marcus, Numberfields. New York: Springer-Verleag, 1970. Moreover, Theorem 1 is generalfor constellations over any number ring R. In this case, the Diophantinenumber φ has to be chosen such that {1, φ, . . . , φ^(M−1)} arealgebraically independent over the number ring considered. Such ageneralization can be useful for including some constellations ofparticular interest. For example, the most energy efficient 8-QAMconstellation is given by{1+i,−1+i,1−i,−1−i,1+√{square root over (3)},−(1+√{square root over(3)}), (1+√{square root over (3)})i,−(1+√{square root over(3)})i}⊂Z[i]∪Z[j].Thus, choosing φ such that {1, φ, . . . , φ^(M−1)} are independent overZ[i]∪Z[j] gives full diversity TAST constellations over the 8-QAMconstellation.

One can also use the new constellations to gain further insight into thetradeoff between performance and complexity for the orthogonal designs.Recently, a framework for the construction of delay-optimal orthogonalST signals was presented. It is easy to see that these signals can beobtained from the construction in (27) if we allow for a slightly moregeneral version of repetition coding where conjugation and/ormultiplication by a constant is allowed for any number of entries. Forexample, for M=4, the delay optimal orthogonal constellation is given by

$\begin{matrix}{O_{4}\overset{\Delta}{=}\begin{pmatrix}u_{1} & u_{2} & u_{3} & 0 \\{- u_{2}^{*}} & u_{1\;}^{*} & 0 & {- u_{3}} \\{- u_{3}^{*}} & 0 & u_{1}^{*} & u_{2} \\0 & u_{3}^{*} & {- u_{2}^{*}} & u_{1}\end{pmatrix}} & (29)\end{matrix}$where u₁, u₂, u₃ belong to the constellation considered. One can simplyidentify the threaded structure in (32) where a full diversitygeneralized repetition code is assigned to each thread. In order toensure orthogonality, however, the fourth thread is left empty. Theempty thread results in a reduced transmission rate and increased PAR.For a fixed throughput, the reduced rate of the orthogonal constellationtranslates into a loss in the coding gain. For example, at a rate of 3bits PCU, the constellation O₄ uses a 16-QAM constellation whereas theconstellation D_(4,4,1) uses an 8-QAM constellation. This results in acoding gain of 2.2185 dB in favor of the D_(4,4,1) code. In addition,the constellation O₄ has a PAR of 12/5, whereas the constellationD_(4,4,1) has a PAR of 5/3 (a gain of 1.5836 dB) in this same scenario.This example illustrates the loss in performance needed to facilitatelinear complexity ML decoding (with the exception of the 2×1 MISOchannel, where the Alamouti scheme is optimal).

One can also generalize this argument to the case of the nondelay-optimal orthogonal signals of rates ½ by considering them as aconcatenation of two delay-optimal threaded constellations. Thisgeneralization, however, does not contribute more insights, and hence,the corresponding details will be omitted for brevity.

E. Extension to MIMO Channel

Now, we extend our approach to MIMO channels (N>1). In this case,sending L=min(M,N) symbols PCU gives the maximum possible rate with apolynomial complexity ML decoding; therefore, the number of informationsymbols to be sent over M transmit antennas and M symbol periods (i.e.,optimal delay) should be M min(M,N). In our approach, we partition theinput information symbols into M streams of L symbols (i.e., u_(j)Δ(u_(j1), . . . , u_(jL))^(T), j=1, . . . , M). Each stream u_(j) isthen fed to a component encoder γ_(j), where the number of coded symbolsat the output of the encoder is M. The output stream from each encoderwill be assigned to a different thread. The component encoders should beconstructed to ensure full diversity in the absence of other threads andguarantee that the threads are transparent to each other. Without lossof generality, we will consider the following assignment of space-timecells to the j^(th) thread (with the convention that time indices span[0,M−1]):l _(j)={(└t+j−1┘_(M)+1,t):0≦t<T} for 1≦j≦M,  (30)where └•┘_(M) denotes the mod-M operation. Note that since the number ofthreads is always equal to M, we avoid having periods of notransmission. The component linear encoders, i.e., γj, j=1, . . . , M,are given byγ_(j)(u _(j))=φ_(j−1) s _(j)=φ_(j−1) {tilde over (M)}u _(j)  (31)where φ_(j−1)j=1, . . . , M, are the Diophantine numbers that separatethe different threads, and {tilde over (M)} is an M×L matrix containingthe normalized first L columns of the M×M full diversity rotation matrixM. For the special case when M is divisible by L, the matrix {tilde over(M)} can be obtained in a slightly different way. Rather than deletingthe last columns of the M×M full diversity matrix, one can obtain {tildeover (M)} by stacking M/L full diversity matrices of dimension L×L. Inthis way, we decrease the algebraic degrees of the rotation matrixelements, and hence, reduce the degrees of the algebraic Diophantinenumbers that achieve full diversity (see Theorem 3).The following examples illustrate the proposed construction.

EXAMPLES 1. M Divisible by L=min(N,M)

For L=N=1, the proposed constellation reduces to that given by (27). Inthis case, {tilde over (M)}=(1, . . . , 1)^(T), andγ_(j)(u_(j))=φ^((j−1)/M){tilde over (M)}u_(j) are the full diversitycomponent encoders. For M=4, L=N=2, we have

$\begin{matrix}{{D_{4,4,2}(u)}\overset{\Delta}{=}{\frac{1}{\sqrt{4}}\begin{pmatrix}s_{11} & {\phi^{3/4}s_{42}} & {\phi^{2/4}s_{31}} & {\phi^{1/4}s_{22}} \\{\phi^{1/4}s_{21}} & s_{12} & {\phi^{3/4}s_{41}} & {\phi^{2/4}s_{32}} \\{\phi^{2/4}s_{31}} & {\phi^{1/4}s_{22}} & s_{11} & {\phi^{3/4}s_{42}} \\{\phi^{3/4}s_{41}} & {\phi^{2/4}s_{32}} & {\phi^{1/4}s_{21}} & s_{12}\end{pmatrix}}} & (32)\end{matrix}$where (S_(k1),s_(k2))^(T)=M(u_(k1),u_(k2))^(T), k=1, . . . , 4, with Mthe optimal 2×2 complex or real full diversity rotation, andu_(k1),u_(k2) εZ [i], k=1, . . . , 4. One proves that φ=e^(iπ/16) (ofdegree 4 over the field containing the 2×2 optimal complex rotation)achieves full diversity over all QAM constellations. Moreover, we havefound the Diophantine number φ=e^(2iπ/7) achieves a local optimum of thecoding gain for the 4-QAM constellation in this configuration. Note thatthe benefit of using a repetition code when L divides M is the smalldegree of the algebraic number field that contains the rotation elementsQ(θ), which has a degree of 4 here as opposed to 8 when using theoptimal 4×4 complex rotation matrix. This implies a smaller degree ofthe Diophantine number φ that separates the different threads, giving inturn a better coding gain.

2. When M is Not Divisible by L=N

For M=3 and L=N=2, we have

$\begin{matrix}{{D_{3,3,2}(u)}\overset{\Delta}{=}{\frac{1}{\sqrt{3}}\begin{pmatrix}s_{11} & {\phi^{2/3}s_{32}} & {\phi^{1/3}s_{23}} \\{\phi^{1/3}s_{21}} & s_{12} & {\phi^{2/3}s_{33}} \\{\phi^{2/3}s_{31}} & {\phi^{1/3}s_{22}} & s_{13}\end{pmatrix}}} & (33)\end{matrix}$where (S_(k2),s_(k2),s_(k3))^(T)=M(u_(k1),u_(k2),0)^(T), k=1, 2, 3, M isthe optimal 3×3 complex or real rotation, u_(k1),u_(k2), k=1, 2, 3,belong to the considered constellation, and φ is chosen to ensure fulldiversity (as formalized in Theorem 3). For example, we have foundφ=e^(2iπ/9) to give a local optima of the coding gain for the 4-QAMconstellation when using the optimal 3×3 real rotation.

The desirable properties of the proposed constellations are formalizedin the following theorem.

Theorem 3. If the Diophantine numbers {φ₀=1, φ₁=φ^(1/M), . . . ,φ_(M−1)=φ^((M-1)/M)} are selected to be transcendental or algebraic suchthat {1, φ, φ², . . . , φ^(M−1)} are independent over the algebraicnumber field Q(θ) that contains the rotation elements and the inputconstellation points, then the new ST constellation will achieve fulldiversity. The PAR of the proposed constellations increases onlylinearly with the number of symbols L (i.e., PAR_(D) _(M,M,L)≦L×PAR_(U)). Moreover, the proposed constellations achieve the optimalEuclidean distance of

$\frac{d_{U}^{2}}{L}.$

Interestingly, if we allow for the use of generalized repetition codes,then we can obtain D_(M,M,L) as the sum of L different variants ofD_(M,M,1). For example, with M=N=L=2, we have

$\begin{matrix}{{D_{2,2,2}(u)} = {\frac{1}{\sqrt{4}} \times {\{ {\begin{pmatrix}u_{1} & {\phi^{1/2}u_{2}} \\{\phi^{1/2}u_{2}} & u_{1}\end{pmatrix} + \begin{pmatrix}u_{3} & {{- \phi^{1/2}}u_{4}} \\{\phi^{1/2}u_{4}} & {- u_{3}}\end{pmatrix}} \}.}}} & (34)\end{matrix}$

We also note that in order to use the proposed constructions with PSKinput constellations, one needs to construct full diversity algebraicrotations for these constellations (i.e., to construct full diversityrotations over Z[w_(M)]). Although previous techniques have beenoptimized for constellations carved from Z[i], one can still utilizethem to construct full diversity rotations over M-PSK by considering theGalois extension of degree M over Q(w_(M)). For example, for M=2 therotation

$M = {\frac{1}{\sqrt{2}}\begin{pmatrix}1 & \phi \\1 & {- \phi}\end{pmatrix}}$guarantees full diversity for two dimensional M-PSK constellations if φis not a quadratic residue in Z[w_(M)] as reported in Table A.E. Trading Diversity for Reduced PAR

Here, we impose the constraint that the PAR of the linear STconstellation is equal to that of the input constellation U. Supposethat the transmitter wants to send L≦min(M,N) symbols, drawn from U, PCUwithout increasing the size of U. Then, as predicted by Proposition 2,the maximum achievable transmit diversity in this case equals

$d = {\lfloor \frac{M}{L} \rfloor.}$Therefore, it suffices to consider only signaling schemes with aninteger L such that M is divisible by L. In the proposed scheme, we onlysend the first MIL columns of the constellation matrix D_(M,M,1)(u) in(27). It is easily seen that this constellation supports L=MN/d symbolsPCU, and achieves a diversity advantage d, while preserving the PAR ofthe input constellation.F. Trading Diversity for Reduced Complexity

We consider the scenario where a sphere decoder with L_(c) complexdimensions is used at the receiver. Given a diversity order of d(divisible by N), let

$T\overset{\Delta}{=}{\frac{d}{N} \leq {M.}}$Then, we construct an M×T TAST constellation with M threads of length Teach. Consider the threading in (36), where we assign scaled fulldiversity diagonal algebraic space-time (DAST) constellations of lengthT to the different threads. To prove that this TAST constellationachieves full diversity (with the correct choice of the Diophantinenumber), we distinguish between two cases:

-   1) If T=M, then one has a square M×M TAST constellation that    achieves full diversity when the Diophantine number φ is chosen to    be algebraic or transcendental satisfying that {1, φ, . . . ,    φ^(M−1)} are algebraically independent over Q(θ), the number field    containing U and the rotation entries.-   2) If T<M, then to prove that this M×T matrix is full rank, we    complete the constellation matrix to a square matrix by adding to    thread l_(j), the numbers φ^((j−1)/M)α_(j,1),φ^((j−1)/M)α_(j,2), . .    . , φ^((j−1)/M)α_(j,M-t), with α_(j,k)≠0εQ(θ), k=1, . . . M−T, j=1,    . . . , M. One can prove that the resulting square matrix satisfies    the full rank condition with the appropriate choice of the    Diophantine numbers as in the first case above. It follows that the    first T columns of this matrix are linearly independent, and    therefore, the considered constellation achieves a diversity of    d=TN.

One can also see that a rate of M symbols PCU is realized by theproposed constellations if a full rate (i.e., one symbol PCU) DASTconstellation is used in each thread. Limiting the dimensionality of thesphere decoder to L_(c) complex dimensions is, however, achieved byzero-setting some of the symbols. So if L_(c)/T≦min(M,N) one can writeTM=L_(c)+n₁T+n₂, with n₁≧0 and 0≦n₂<T. This suggests that if one deletesthe last n₁ threads in our M×T TAST constellation, and sets u_(M−n) ₁_(+1,T−n) ₂ ₊₁= . . . =u_(M−n) ₁ _(+1,T)=0, one obtains a transmissionrate of L_(c)/T symbols PCU while allowing for the polynomial complexitysphere decoder with L_(c) dimensions. Finally, we note that deletingsome threads and zero-setting some symbols in a thread does not affectthe diversity gain.

The following examples illustrate the proposed scheme. Let M=3, N=3, andconsider the following choices of L_(c).

1) L_(c)=min(M,N)=3 allows for the following choices of (d,L):

-   -   (a) (d=3, L=3): The truncated M×d TAST constellation reduces to        the well-known V-BLAST system.    -   (b) (d=6, L=3/2): One sends the following TAST constellation

$\begin{matrix}{\frac{1}{\sqrt{3}}\begin{pmatrix}s_{11} & {\phi^{2/3}s_{32}} \\{\phi^{1/3}s_{21}} & s_{12} \\{\phi^{2/3}s_{31}} & {\phi^{1/3}s_{22}}\end{pmatrix}} & (35)\end{matrix}$

-   -   (c) where (s_(j1),s_(j2))^(T)=M(u_(j1),u_(j2))^(T),        u_(j1),u_(j2)εU,j=1, 2, 3, with M the 2×2 optimal rotation        matrix, and φ chosen such that {1, φ, φ²} are independent over        Q(θ) the field containing U and the rotation entries (e.g., when        U is a QAM constellation, and M is complex, then θ=e^(iπ/4) and        one can choose φ=e^(2iπ/5) of degree 4 over Q(θ) to ensure a        diversity d). To obtain a rate of 3/2 compatible with the        complexity L_(c)=3, one sets u₃₁=u₃₂=u₂₁=0.    -   (d) (d=9, L=1): This is achieved by the TAST constellation in        (27).        2) L_(c)=6 allows the following two possibilities for (d,L)        (24).    -   (e) (d=6, L=3): This is achieved by the constellation in (48)        without zero-setting any information symbols.    -   (f) (d=9, L=2): This is achieved by the constellation in (42)        where (s_(j1),s_(j2),s_(j3))_(T)=M(u_(j1),u_(j2),u_(j3))^(T),        u_(j1)u_(j2),u_(j3)εU, j=1, 2, the third thread is deleted, and        φ chosen such that {1, φ} are independent over Q(θ) associated        with rotation M.        3) L_(c)=9 allows for transmitting at full rate and full        diversity by using the TAST constellation

$\begin{matrix}{\frac{1}{\sqrt{3}}\begin{pmatrix}s_{11} & {\phi^{2/3}s_{32}} & {\phi^{1/3}s_{23}} \\{\phi^{1/3}s_{21}} & s_{12} & {\phi^{2/3}s_{33}} \\{\phi^{2/3}s_{31}} & {\phi^{1/3}s_{22}} & s_{31}\end{pmatrix}} & (36)\end{matrix}$where (s_(j1),s_(j2),s_(j3))^(T)=M(u_(j1),u_(j2),u_(j3))^(T),u_(j1),u_(j2),u_(j3)εU, j=1, 2, 3, with M the 3×3 optimal rotationmatrix, and φ is chosen such that {1, φ, φ²} are independent over Q(θ)associated with rotation M.

One can use the same technique for optimizing the performance of theproposed constellations with the nulling and cancellation receiver(i.e., by finding the optimal pair of diversity and number of symbolsPCU). Although this approach is motivated by a conjecture, the numericalresults in the following section demonstrate the significant gains,compared to the V-BLAST for example, in various scenarios.

TABLE II Comparisons Of The Mutual Information Of The TASTConstellations and The Linear Dispersion Codes at an SNR of 20 dB TASTconst., LD const., Channel (M, N) T = M T[2, Table I] Capacity (2, 1)5.893 (6.28, T = 2) 6.28 (3, 1) 5.893  (6.25, T = 4), 6.41 (6.28, T = 6)(3, 2) 11.27 (11.63, T = 4)  12.14 (4, 1) 5.893 (6.34, T = 4) 6.47 (4,2) 11.27 (11.84, T = 6)  12.49 (8, 4) 22.14 (23.10, T = 8)  24.94

Specific representations of the encoders and codes have been presentedthat, for example involve linear transformations followed by discreteFourier transformations. More generally, any of the codes defined hereincan be thought of as defining a set of constraints that will determinethe output from the input. The same set of constraints may beimplemented by a different physical encoder design. For the purpose ofthis description, an encoder is “equivalent” to another encoder if the“black box” transfer function between the input and the output of thetwo encoders is the same.

Numerous modifications and variations of the present invention arepossible in light of the above teachings. It is therefore to beunderstood that within the scope of the appended claims, the inventionmay be practised otherwise than as specifically described herein.

1. A transmitter comprising: M antennas; and a space-time coder, thespace-time coder including: linear encoders configured to perform arespective full diversity linear encoding on each of P sets of Kmodulated symbols to generate P sets of T linearly encoded symbols; anda threader configured to assign each one of the P sets of T linearlyencoded symbols to a corresponding one of threads of a M×T dimensionalspace-time code, the threads being allocated such that only one of the Mantennas is used for transmission at each time interval of a given oneof the threads and over time the M antennas are used substantiallyequally likely, wherein M>=2, 2<=P<=M, and T>=M, and wherein: each oneof the linear encoding performed on each of the P sets of K modulatedsymbols is of full diversity and different from one another; and the M×Tdimensional space-time code is of full diversity.
 2. The transmitter ofclaim 1, wherein each one of the linear encoding performed on each ofthe P sets of K modulated symbols has a non-zero minimum productdistance.
 3. The transmitter of claim 1, wherein the linear encoders areconfigured to perform a respective one of linear transformations on eachof the P sets of K modulated symbols to generate P sets of T linearlytransformed symbols and to apply a respective one of phase rotations toeach of the P sets of T linearly transformed symbols to generate the Psets of T linearly encoded symbols.
 4. The transmitter of claim 3,wherein the phase rotations are different for each of the P sets of theT linearly transformed symbols.
 5. The transmitter of claim 3, whereinthe linear transformations are identical and unitary.
 6. The transmitterof claim 3, wherein the linear transformations comprise at least in parta discrete Fourier transform (DFT).
 7. The transmitter of claim 3,wherein the linear transformations comprise a product of a discreteFourier transform (DFT) and a diagonal matrix.
 8. The transmitter ofclaim 1, wherein the P sets of K modulated symbols are modulatedaccording to quadrature amplitude modulation (QAM), pulse amplitudemodulation (PAM), or phase shift keying (PSK).
 9. A method comprising:performing a respective full diversity linear encoding on each of P setsof K modulated symbols to generate P sets of T linearly encoded symbols;and performing a threading operation to assign each one of the P sets ofT linearly encoded symbols to a corresponding one of threads of a M×Tdimensional space-time code; and transmitting the M×T dimensionalspace-time code using M antennas, the threads being allocated such thatonly one of the M antennas is used for transmission at each timeinterval of a given one of the threads and over time the M antennas areused substantially equally likely, wherein M>=2, 2<=P<=M, and T>=M, andwherein: each one of the linear encoding performed on each of the P setsof K modulated symbols is of full diversity and different from oneanother; and the M×T dimensional space-time code is of full diversity.10. The method of claim 9, wherein each one of the linear encodingperformed on each of the P sets of K modulated symbols has a non-zerominimum product distance.
 11. The method of claim 9, wherein performinga respective full diversity linear encoding on each of P sets of Kmodulated symbols comprises: performing a respective one of lineartransformations on each of the P sets of K modulated symbols to generateP sets of T linearly transformed symbols; and applying a respective oneof phase rotations to each of the P sets of T linearly transformedsymbols to generate the P sets of T linearly encoded symbols.
 12. Themethod of claim 11, wherein the phase rotations are different for eachof the P sets of the T linearly transformed symbols.
 13. The method ofclaim 11, wherein the linear transformations are identical and unitary.14. The method of claim 11, wherein the linear transformations compriseat least in part a discrete Fourier transform (DFT).
 15. The method ofclaim 11, wherein the linear transformations comprise a product of adiscrete Fourier transform (DFT) and a diagonal matrix.
 16. The methodof claim 9, wherein the P sets of K modulated symbols are modulatedaccording to quadrature amplitude modulation (QAM), pulse amplitudemodulation (PAM), or phase shift keying (PSK).